Pairwise Stone space

In mathematics and particularly in topology, a pairwise Stone space is a bitopological space $(X,{\tau }_1,{\tau }_2)$ that is pairwise compact, pairwise Hausdorff, and pairwise zero-dimensional.

Pairwise Stone spaces are a bitopological version of Stone spaces.

Pairwise Stone spaces are closely related to spectral spaces.

Theorem: If $(X,\tau )$ is a spectral space, then $(X,\tau ,{\tau }^{\ast })$ is a pairwise Stone space, where ${\tau }^{\ast }$ is the de Groot dual topology of $\tau$ . Conversely, if $(X,{\tau }_1,{\tau }_2)$ is a pairwise Stone space, then both $(X,{\tau }_1)$ and $(X,{\tau }_2)$ are spectral spaces.

See also

• Bitopological space
• Duality theory for distributive lattices